Discrete Pseudo Almost Periodic Solutions for Some Difference Equations

Elhadi Ait Dads, Khalil Ezzinbi, Lahcen Lhachimi

.
University Cadi Ayyad, Faculty of Sciences Semlalia, Department of Mathematics, Marrakesh, Morocco.

DOI: 10.4236/apm.2011.14024                

Abstract

In this work, we study the existence and uniqueness of pseudo almost periodic solutions for some difference equations. Firstly, we investigate the spectrum of the shift operator on the space of pseudo almost periodic sequences to show the main results of this work. For the illustration, some applications are provided for a second order differential equation with piecewise constant arguments.

Keywords

Difference Equations, Pseudo Almost Periodic Sequences, Schift Operator

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	C. Corduneanu, “Almost Periodic Discrete Processes,” Libertas Mathematica, Vol. 2, 1982, pp. 159-169.
[2]	J. Hong and C. Nú?ez, “The Almost Periodic Type Difference Equations,” Mathematical and Computer Modelling, Vol. 28, No. 12, 1998, pp. 21-31. doi:10.1016/S0895-7177(98)00171-X
[3]	S. Elaydi, “An Introduction to Difference Equations,” 3rd Edition, Springer-Verlag, Berlin, 2000.
[4]	E. A. Dads and L. Lha-chimi, “New Approach for the Existence of Pseudo Almost Periodic Solutions for Some Second Order Differential Equa-tion with Piecewise Constant Argument,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 64, No. 6, 2006, pp. 1307-1324. doi:10.1016/j.na.2005.06.037
[5]	A. I. Alonso, J. Hong and J. Rojo, “A Class of Ergodic Solutions of Differentiale Quations with Piecewise Constant Arguments,” Dynamic Systems and Applications, Vol. 7, 1998, pp. 561-574.
[6]	A. M. Fink, “Almost-Periodic Differential Equations, Lecture Notes in Mathematics,” Springer-Verlag, Berlin, Vol. 377, 1974.
[7]	S. Zaidman, “Solutions Presque-Périodiques des équations Dif-férentielles Abstraites,” L’Enseignement Mathé- matique, Vol. 24, No. 1-2, 1978, pp. 87-110.
[8]	S. Zaidman, “A Non-Linear Abstract Differential Equation with Al-most-Periodic Solution,” Rivista di Matematica della Univer-sità di Parma, Vol. 10, No. 4, 1984, pp. 331-336.
[9]	C. Zhang, “Pseudo Almost Periodic Functions and Their Applica-tions,” Ph.D Thesis, University of Western Ontario, London, Canada, 1992.
[10]	C. Zhang, “Pseudo Almost-Periodic Solu-tions of Some Differential Equations,” Journal of Mathemati-cal Analysis and Applications, Vol. 181, No. 1, 1994, pp. 62-76. doi:10.1006/jmaa.1994.1005
[11]	C. Zhang, “Integration of Vector-Valued Pseudo Almost Periodic Functions,” Proceeding of the American Mathematical Society, Vol. 121, No. 1, 1994.
[12]	C. Zhang, “A Characterization of Pseudo Almost Periodic Functions in Fourier Analysis,” Acta Analysis Func-tionalis Applicata, Vol. 4, 2002, pp. 110-114.
[13]	C. Zhang, “Almost Periodic Type Functions and Ergodicity,” Kluwer Academic Publishers, Dordrecht, 2003.
[14]	S. M. Shah and J. Weiner, “Advanced Differential Equations with Piecewise Constant Argument Deviations,” International Journal of Mathematics and Mathematical Sciences, Vol. 6, No. 4, 1983, pp. 671-703. doi:10.1155/S0161171283000599
[15]	Y. Rong and H. Jialin, “The Existence of Almost Periodic Solutions for a Class of differential Equations with Piecewise Constant Argument,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 28, No 8, 1997, pp. 1439-1450. doi:10.1016/0362-546X(95)00225-K
[16]	Y. Rong, “Existence of Almost Periodic Solutions of Second Order Neutral Delay Differential Equations with Piecewise Constant Argument,” Science in China (Series A), Vol. 41, No. 3, 1998, pp. 232-241. doi:10.1007/BF02879041
[17]	R. Yuan, “Pseudo-Almost Pe-riodic Solutions of Second-Order Neutral Delay Differential Equations with Piece- wise Constant Argument,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 41, No. 7-8, 2000, pp. 871-890. doi:10.1016/S0362-546X(98)00316-2
[18]	Y. Rong and T. Kupper, “On Quasi-Periodic Solutions of Differential Equa-tions with Piecewise Constant Argument,” Journal of Mathe-matical Analysis and Applications, Vol. 267, No. 1, 2002, pp. 173-193. doi:10.1006/jmaa.2001.7761
[19]	Y. Rong, “On a New Al-most Periodic Type Solution of a Class of Singularly Perturbed Differential Equations with Piecewise Constant Argument,” Science in China (Series A), Vol. 45, No. 4, 2002, pp. 484-502. doi:10.1007/BF02872337
[20]	Y. Rong, “The Existence of Almost Periodic Solutions of Retarded Differential Equations with Piecewise Constant Argument,” Nonlinear Analysis: The-ory, Methods and Applications, Vol. 48, No. 7, 2002, pp. 1013-1032